Related papers: On Hodge Structures and Periods -- Part I: Algebra…

In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…

We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple…

We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…

This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal…

We investigate the structures of Hopf $\ast$-algebra on the Radford algebras over $\mathbb {C}$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.

It is known that the algebraic \deRham cohomology group $\hDR{i}(X_0/\Q)$ of a nonsingular variety $X_0/\Q$ has the same rank as the rational singular cohomology group $\h^i\sing(\Xh;\Q)$ of the complex manifold $\Xh$ associated to the base…

This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own…

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and…

We study the relationships among the various forms of the $q$ oscillator algebra and consider the conditions under which it supports a Hopf structure. We also present a generalization of this algebra together with its corresponding Hopf…

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.

This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index…

In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations…

We show that reasonably well behaved 3d and 4D TQFts must contain certain algebraic structures. In 4D, we find both Hopf categories and trialgebras.

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic…

This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…

We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at…

Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order…